Introducing robotic origami folding

Introducing robotic origami folding
Devin J. Balkcom and Matthew T. Mason
Abstract— Origami, the human art of paper sculpture,
is a fresh challenge for the field of robotic manipulation,
and provides a concrete example for many difficult and
general manipulation problems. This paper presents some
initial results, including the world’s first origami-folding
robot, definition of a simple class of origami for which we
have designed a complete automatic planner, an analysis of
the kinematics of more complicated folds, and some new
theorems about foldability.
I. I NTRODUCTION
The design and manipulation of foldable objects like
paper bags, newspapers, fast-food containers, tents, mirrors for space telescopes, cardboard boxes, and clothing
present some challenges; we hope that exploring origami
folding will lead to some new insights into these hard
problems. The primary contribution of this paper is the
definition of a new task domain; although origami has
been widely studied in the mathematics community, this
appears to be the first examination of origami from the
robotic manipulation perspective. The results presented
fall in three classes: designing for foldability, folding
manipulation, and analysis of closed chains.
There is a rich field of work on origami design in the
mathematics community; Demaine et al. [1] provides a
survey. Robert Lang’s papers and TreeMaker software [7]
are usually credited with being the first in-depth work.
Our contributions to origami design are the extension
of some well-known necessary conditions for origami
patterns to be flat-foldable. Work by Hull [5] provides a
good survey, and is a starting point for our results.
Our primary interest in origami is manipulation. Much
can be learned from human origami experts. One observation is that most origami folds are quite simple.
Folds are introduced sequentially, and previously folded
facets are ‘locked’ down, allowing the origami to be
treated as a serial chain. We define two basic folding
techniques, and present a complete planner for these
types of folding. Our folds are similar to those used
by the virtual origami software described in Miyazaki et
al. [14]; our contribution relative to this work is a careful
analysis of the conditions under which these folds can be
made, a requirement for automatic planning. We present
a paper airplane and a hat that are simply foldable, and
show that the traditional ‘samurai hat’ design can be
folded using eight simple folds and a single book fold,
without flipping the paper. We present a prototype of a
machine that can fold simply-foldable origami – a movie
is available at http://www.cs.cmu.edu/˜devin.
Fig. 1.
A machine that can fold simple origami.
Box folding and sheet metal bending are perhaps
the two applications in robotic manipulation closest
to origami folding. Lu and Akella ([10],[11]) describe
a planner (modelled on traditional planners for robot
arms) that enumerates all collision-free sequences to fold
a carton blank into a carton. Possibly the most complete
and practical system for sheet metal bending is that
described by Gupta et al [2], consisting of a high-level
planner that determines the sequence of bends, and a
number of low level planners for individual actions.
Unlike sheet metal and cardboard cartons, origami is
often flat when folded, and typically contains closed
chains in its kinematic structure. These characteristics
have implications for collision detection, state representation, and available actions.
Paper facets around a vertex form a spherical closed
chain, and complicated folds like reverse folds and
prayer folds manipulate the paper as a closed chain.
Huffman [4] and McCarthy [12] both solve four-bar
spherical linkages to show the relationship between joint
angles; we extend this work to the case of n-bar linkages.
Although beyond the scope of this paper, planning for
simultaneous manipulation of links in closed chains is a
primary direction of future work; probabalistic methods
of planning for closed chains are described in LaValle et
al. [8] and Han and Amato [3]. Lenhart and Whitesides [9] and Milgram and Trinkle [13], [15] describe
complete and efficient planners for the specific case of
planar n-bar linkages without self-intersections.
II. R EPRESENTING
ORIGAMI
Define the facets of an origami piece to be the unfolded
regions of paper. In the present work, we model only
polygonal facets, and treat them as rigid links. The facet
edges interior to the paper are creases; a crease line is a set
Fig. 2.
The pattern, facet graph, and a facet tree for a samurai hat.
of colinear creases. Define the origami pattern to be the
location of creases on unfolded origami. Creases meet at
interior vertices of the pattern; if n creases meet, we say
that a vertex is of degree n. The angles between creases
around a vertex in the pattern are called sector angles.
Each crease connects two facets, and creases are the
edges of the facet graph. We say that any tree that spans
the facet graph is a facet tree. Facet trees are easy to construct; any complete search method such as breadth-first
or depth-first search is suitable. Creases not contained in
a facet tree will be said to be virtually (but not necessarily
physically) cut relative to that tree.
A facet tree implies a parent-child relationship between two facets connected by a crease. We will choose
the convention that all facets are described by a counterclockwise set of points in the pattern; we will associate
a unit vector with each crease such that the vector’s
direction agrees with the order of vertices in the child
facet. We then describe the crease angle as the angle
between a parent facet and its child; the sign is chosen
to be consistent with the ‘right hand rule’ applied to the
crease vector.
Given a pattern and any facet tree, the crease angles
associated with all uncut creases determine the kinematics of the origami mechanism – the pose of each facet
and the angle of each cut crease can be determined by
traversing the facet tree applying rotations to descendent
facets.
Since origami is flexible and can be folded essentially
flat,
it is convenient to allow crease angles in the range
π, π . Flat origami illustrates a difficulty with a purely
‘kinematic’ model, however. Figure 3 shows the problem;
although in the limit the crease angles of two flat origami
pieces may be the same, the order in which facets are
stacked is important. We will call a group of coplanar
facets a compound facet. With each compound facet we
associate a normal vector and a facet stacking relative
to this vector. The height of a facet is its height in the
stacking, and the height of a crease is the height of its
child facet.
III. N ECESSARY
CONDITIONS FOR FLAT- FOLDABILITY
We say origami is flat if all facets lie in a single plane.
Origami patterns must satisfy some well-known necessary conditions in order to be flat-foldable; for example,
Kawasaki’s theorem, Maekawa’s theorem, and Meguro’s theorem. For discussion and proofs of these theorems, see
Fig. 3.
Two nearly flat origami pieces.
Hull [5]. In this section, we present some extensions of
these ideas based on the definition of the facet tree and
stacking.
Flat origami has an important property: the rigid-body
pose of each facet can be determined solely from the
pattern, without knowledge of stacking or how to fold
the pattern. We can edge fold any pattern by selecting
any facet tree and traversing the tree, completely folding
one uncut crease at each step and allowing facets to
pass through one another where necessary. We say that
an origami piece is ‘edge-foldable’ if, after applying
edge folding, both sides of each cut crease line up. Flat
origami is edge-foldable; this observation may be seen as
a generalization of Kawasaki’s theorem to multi-vertex
patterns.
Maekawa’s theorem provides a necessary condition
on the crease angles around each vertex in the folded
origami. If we know the stacking order for the goal, it is
possible to do better:
Theorem 1: The facet graph and the goal stacking order completely determine which creases are mountain
creases and which are valley creases.
Proof: Edge fold the pattern to find the facet normals
in the goal state. Each crease connects two facets; the sign
of the crease angle depends only on the normals of the
two adjacent facets and the heights of the two facets.
For example, if the normal to the parent facet points up,
and the child facet is above the parent, the crease angle
is positive.
There are further global necessary conditions for
origami to be flat-foldable; the interested reader is referred to Hull [5].
IV. O RIGAMI
SKILLS
What skills do human origami experts use? http://
folds.net lists some skills from simple to advanced.
The first three categories are:
Pureland: valley fold, mountain fold, turn over,
rotate, book fold, . . .
Simple: reverse folds, prayer fold, . . .
Low-Intermediate: squash fold, petal fold, . . .
Our version of the list omits some folds: pleat, radial
pleat, cupboard fold, blintz fold, waterbomb base, preliminary fold, bird base, frog base, rabbit ear, and kite
base. Each omitted fold can be folded by a sequence of
the listed folds; for example, a blintz fold valley folds
four corners of a square into the center. These compound
Fig. 4.
A sequence of two simple folds.
folds do provide an advantage that we ignore in this
paper – they allow landmarking (the center of a square
can be precisely located by making a blintz fold), and
also form larger building blocks for origami instructions.
The most basic skills are those in the ‘Pureland’
origami designed by John Smith, which allows only
mountain and valley (‘land’) folds. (The motivations
for the creation of Pureland origami were two-fold:
the asthetics of minimalist design, and making origami
accessible to handicapped children; see http://www.
users.waitrose.com/˜pureland/.) With the exception of a few three-dimensional manipulations allowed
in Pureland (primarily tucking and inserting of flaps),
the fold models we present below are sufficient to describe Pureland origami.
A. Simple folds
The most basic origami fold takes all paper on one side
of a crease line and folds it to the other side; the final
pose of facets can be determined by reflection across the
crease line. Figure 4 shows a sequence of two simple
folds.
Given a flat origami state, what simple folds are
possible? First, find the minimal set of lines that contains
all creases. Discard any crease lines that cross a facet.
Each remaining crease line divides the facets into two
compound facets; we arbitrarily assign one to be the
‘base’ and one the ‘flap’. During folding, the base will
not move. The flap can be folded either up (a valley simple
fold) or down (a mountain simple fold).
To execute the fold, all creases colinear with the
crease line are folded simultaneously. During folding, the
heights of facets in the flap are reversed, and then either
stacked above or below the base, forming a single new
compound facet.
There are many interesting origami designs that can
be simply folded. Figure 5 shows two of these: a cup,
and a paper airplane.
B. Book folds
Figure 6 shows two examples of origami that cannot
be simply folded. Origami folders often refer to the type
of fold shown in figure 6b as a ‘book fold’ – the moving
facet acts like the page of a book. We will define a book
fold as follows: a book fold reflects a set of facets on one
side of a set of colinear creases onto the other, such that
Fig. 5.
Fig. 6.
Simple foldings of two traditional origami designs.
Two designs that can be book folded but not simply folded.
the continuous rigid body rotation corresponding to the
reflection does not cause self-intersection of the origami
or tear any creases. We will call the moving facets the
flap, and the fixed facets the base. We call the creases
involved in the fold the active creases.
Theorem 2: The set of active creases in a book fold cuts
the facet graph, separating the flap from the base.
Proof: Any crease that connects the base and flap and
is colinear with the crease line must be active; the crease
angle will be the angle between the base and flap, up
to sign. Any crease that connects the base and flap but
is not colinear with the crease line will be torn by any
rigid body rotation of the flap around the crease line.
We can determine the set of all book folds from
an origami state. First, enumerate all crease lines. Sort
the creases by height. Consider all sequential combinations of creases that contain either the minimum- or
maximum-height crease. Test each of these crease sets
to determine if it cuts the facet graph into at least two
pieces. All pieces strictly to the left of the current crease
line are candidate flaps, as are all pieces strictly to the
right. Test each candidate flap to see if and in which
direction(s) folding is possible without self-intersection
of the origami; this can be accomplished by polygon
intersections in the plane of the compound facet.
Simple folding is a special case of book folding, and
execution of a book fold is similar to execution of a
simple fold – reflect the flap across the crease line, flip
flap stacking, and stack the flap either above or below
the base, depending on the direction of the fold.
V. A PLANNER
FOR BOOK - FOLDABLE ORIGAMI
Origami that can be folded using book folds is flat
after each fold. Since the motion of the flap occurs
out of the plane of the base, collision detection is only
Fig. 7.
Automatically planned folding of the samurai hat.
necessary at the beginnings and ends of folds, and only
requires simple polygon intersection tests. Futhermore,
the origami state after each fold is just the stacking of the
facets, and is thus discrete. We have implemented a complete graph search planner for book-foldable origami;
the nodes of the graph are flat origami states. Figure 7
shows an automatically planned folding of the samurai
hat comprised of eight simple folds and one book fold.
The input to the planner is the pattern and the desired
stacking of the facets. We now describe the algorithm.
Use the goal stacking and the pattern to determine signs
on the crease angles. Insert the pattern into the search
queue as the initial state. While the search queue has
elements, pop, test for goal state, and if goal, backchain
to find the plan. Otherwise, determine the book folds
from the state and generate successor states. Cull any
states that have crease angles that do not agree with
those of the goal state. Also cull any states that have
been previously visited. Insert remaining states into the
visited list and into the search queue.
The visited list is implemented as a hash table that
hashes based on the integer heights of facets in the stacking. Before testing against the visited list, the compound
facet of the state is collapsed to determine a minimal
stacking. The algorithm to find the minimal stacking
is essentially a bubble sort – each facet is allowed to
bubble downwards in the stacking as long as it does not
intersect with any facets in the level underneath it.
The planner implementation is about 5000 lines of C++
code, and was run on a 500 mhz Pentium III. The table
below shows results for four traditional origami designs.
Origami
Cup
Airplane
Hat
Samurai hat
creases
9
9
14
20
nodes
30
24
75
4250
time (sec)
.1
.1
.5
110
folds
5
5
5
9
For the samurai hat, more than 99% of the CPU time
was spent in polygon intersections to determine minimal
stackings and to find book folds.
VI. A N
ORIGAMI FOLDING MACHINE
Figures 1 and 8 show a machine designed to allow a
4DOF Adept SCARA robot arm to make simple folds.
A vacuum pad grips the paper, while the arm positions
the paper over the folding device. The design is based
Fig. 8.
Fig. 9.
A machine that can fold simple origami.
Making a simple fold with the machine; side view.
on the following observation: any origami piece that can
be folded by a sequence of flips and simple folds can be
folded by a single initial flip and a sequence of valley
simple folds.
The folding procedure is outlined in figure 9. The
paper is first positioned on the table above a slot; the
position of the paper determines the location of the
crease. A blade presses the paper into the slot (step
2); friction holds the paper in the slot as the blade
is removed. The slot clamps shut, forming the crease
(step 3). Steps 4 and 5 show a method for removing
the paper from the slot and placing it flat on the table;
this is required since the arm used only provides one
rotational degree of freedom at the wrist. First the blade
sweeps across the paper, forcing it to lie flat. The clamp
is released while the blade holds the paper against the
table; the springiness of the paper allows it to swing free
of the slot.
The primary goal of our design was to demonstrate
that origami could be folded by machine. Error is inherent in the frictional insertion of the paper into the
slot, and in the positioning and grasping of the paper,
which are all open-loop. Typically, there is an error on the
order of one millimeter in the positioning of each crease.
Accumulation of errors limits the number of folds that
Fig. 10.
Only the first design is book-foldable.
can be made reliably; nonetheless, the machine is able to
completely fold both the simple airplane design shown
in figure 5 and a simplified fourteen-crease version of
the samuarai hat – each design takes about two minutes
to fold.
VII. C LOSED
CHAIN FOLDING
Many origami designs cannot be book folded; figure 10 shows some examples. Inside and outside reverse
folds, squash folds, and petal folds all manipulate four
creases simultaneously.
The planner for book folding described above relies
on a key observation – self-intersection can occur only
when the origami is flat. It turns out that a similar result
holds for origami where four creases that intersect at a
vertex are manipulated simultaneously:
Theorem 3: Rigid flat-foldable degree-four origami can
only self-intersect when flat.
Proof: (sketch) First show that continous motions of
the origami cannot cause self-intersection without at
least one crease angle reaching either zero or 2π. To
complete the proof, consider the cases under which a
crease angle can become flat. Huffman [4] derives a
relationship between opposite crease angles m and n for
degree-four origami,
sin A sin B 1 cos m ,
(1)
sin C sin D
where A, B, C, and D are sector angles counter-clockwise
from the crease corresponding to m. Kawasaki’s theorem
constrains A, B, C, and D; for flat-foldable origami,
n m. From this follows that if no two adjacent sector
angles are equal or sum to π, and if one crease angle is
zero or π, all of the others must be zero or π as well; the
origami is flat. If two adjacent sector angles are equal or
sum to π, book folding may be possible, but the theorem
still holds.
Prayer folding and more advanced origami skills require the simultaneous manipulation of more than four
creases. In the remainder of the section, we present
the relationship between crease angles for vertices of
arbitrary degree; our result is also applicable to the case
where there is missing or excess sector angle around the
vertex.
The mobility of a vertex
of degree n is n 3. We
will therefore choose n 3 arbitrary independent crease
1
cos n
Fig. 11.
Solving for three dependent crease angles.
angles as input, and solve for the remaining crease
angles. (In the special case where the dependent crease
angles are sequential, a simpler solution is possible using
a reverse kinematics approach similar to that described
in [3].)
Figure 11 shows the procedure; φ1 , φ2 , and φ3 are the
crease angles to be solved for. First cut the crease corresponding to φ3 . Anchor the next (counter-clockwise)
facet, and choose a coordinate system so that the φ1
crease falls on the x axis. The closure constraint can be
written using a product of rotation matrices,
R1 R x φ1 R2 Rz α R x φ2 Rz
α R3 p l
pr ,
(2)
where Ri are rotation matrices corresponding to independent crease angles, and p l and pr are points that must
touch to close the loop. Rewrite equation 2:
R x φ1 ZR x φ2 a
(3)
b,
where
Z
R2 R z α (4)
a
Rz
(5)
R1T pr
b
α R3 p l
(6)
are known. Multiplying out equation 3 gives three equations, the first of which is
k3
k1 cos φ2 k2 sin φ2 ,
(7)
with known variables
k1
k2
k3
z13 a3
(8)
z13 a2 z12 a3
b1 z11 a1 .
(9)
(10)
z12 a2
If k1 k2 0, then equation 7 implies that φ2 can take
on any value. Otherwise, equation 7 has the solution(s)
φ2
atan k2 , k1 acos
k3
k21 k22
.
(11)
There may be zero, one, two, or infinitely many solutions
for φ2 . For each value of φ2 , the remaining two rows
Fig. 12.
A multi-vertex pattern with a mobility of one.
of equation 3 can be used to solve for φ1 , which either
has a unique value or is unconstrained. φ3 is uniquely
determined by the angle between the normals to the
facets at either end of the cut chain.
VIII. C ONCLUSION
AND
A CKNOWLEDGMENTS
We have presented an articulated-rigid-body model
that is rich enough to allow flat folding of origami.
We have explored aspects of common skills used to
fold origami, including mountain and valley folding,
book folding (colinear creases), reverse folding, squash
folding, petal folding (degree-four vertices), and prayer
folding (high-degree vertices).
Future work includes an exploration of the ‘low-level’
manipulation skills (like landmarking) that humans use
to precisely and reliably place creases in flexible paper.
We also intend to explore further aspects of complex
closed-chain manipulation. The configuration spaces of
closed chains can be complicated, and understanding
connectedness of these c-spaces is a very interesting
problem; we have had some initial success in this direction using techniques based on those described in [13].
Another interesting problem is understanding multivertex patterns like that shown in figure 12. Huffman [4]
points out that by viewing the vertices as nodes and
the creases as edges, it is possible to find a relationship
between crease angles for the network shown. The relationship between crease angles described in section VII
allows this procedure to be extended to multi-vertex
patterns with vertices of degree higher than four; in this
case, there may be many ‘input crease angles’ in the
network. Understanding the singularities and connectedness of c-spaces for these compound closed chains seems
to be a challenging problem.
A final interesting result is that common paper shopping bags cannot be folded without bending the facets –
our simple model of origami does not apply to what is
probably the most common origami structure! Although
we omit the proof, the open and folded configurations
of the bag turn out to be isolated points in c-space.
This could be considered a design feature; shopping
bags tend to stay open since facets resist being bent.
However, arbitrary out-of-plane flexing of paper facets
is not possible, since paper does not stretch. (See work
on developable surfaces and properties of paper; Kergosien et al [6] provides the most complete survey.)
An interesting question is whether facets must crinkle
as well as bend during folding of the shopping bag.
In current work with Demaine and Demaine, we have
shown that a shopping bag can be flattened by adding a
finite number of additional creases; this result gives an
important boundary for the well-known bellows theorem,
which states that closed flexible polyhedra have constant
volume.
Thanks to Doug James, James Kuffner, Jeffry C. Trinkle, R. James Milgram, Ravi Balasubramanian, Siddhartha Srinivasa, and the members of the CMU Center
for Foundations of Robotics for their helpful comments
and insights. The work of Devin Balkcom was supported
by a Department of Energy Computational Science graduate fellowship. Funding was further provided by NSF
grants IIS 0082339 and IIS 0222875.
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